Suppose that T : R° → R² is a linear transformation whose standard matrix has 2 pivots.Is it possible to find two different vectors u, v such that T(u) =T(v)? That is, is(i)T one-to-one? Answer and explain briefly how you know.(ii)Given any vector b in R?, is it possible to find x such that T(x) = b? That is,does T map onto R?. Answer and explain briefly how you know.

Answered: Image /qna-images/answer/7ee9c900-2fb7-4a75-9130-9cb45dffe26a.jpg

Answered: Suppose that T : R' → R' is a linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider matrix A = Answer: 0 1 (₁2) G -1 2 and vectors x = (¹2₂) and y = (1). . Calculate matrix B = (A – AT)² and vector w = Bx. Then, evaluate the scalar product w · y. [Please enter your answer numerically. You will be marked correct as long as what you enter is within 0.5 of the correct answer.]
[10] Find the standard matrix representation of the linear transformation from R? into R² that corresponds to reflection in the line y = x followed by projection onto the y-axis.
find the standard matrix for the reflection of IR squared about the line that makes an angle of π/4 with the positive x-axis, and then use that matrix to find the reflection of (1,2) into this line.
For a 4×4 matrix whose top three rows are arbitrary and whose bottom row is (0, 0, 0, 1), show that the points (x, y, z, 1) and (hx, hy, hz, h) transform to the same point after homogenization.
Find a preimage of the vector 4 V= under the linear transformation T given by the matrix -3] = [44] -4 14 That is find the vector u such that T(u) = v. A U₁ ū₂
5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.
5. Find the standard matrix of the given linear transformation: (a) P: R? - R? projects a vector onto the line y = -2x. For the vector u = find P(ü). (b) Rg: R3 - R rotates a vector clockwise about the z - axis through the angle 30°. For the 21 vector u = 2| find R9 (ũ). 6. For the following linear transformations find S • T and T • S. Indicate the domain and codomain of the new linear transformations. x+ y+z] x-y-zl 2x + 7. Find the inverse of the following linear transformation: [3x + 2y] [5x + 3yl %3D
Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…

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